YES

Problem 1: 

(VAR v_NonEmpty:S X:S X1:S X2:S)
(RULES
a__f(0) -> cons(0,f(s(0)))
a__f(s(0)) -> a__f(a__p(s(0)))
a__f(X:S) -> f(X:S)
a__p(s(X:S)) -> mark(X:S)
a__p(X:S) -> p(X:S)
mark(0) -> 0
mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S)
mark(f(X:S)) -> a__f(mark(X:S))
mark(p(X:S)) -> a__p(mark(X:S))
mark(s(X:S)) -> s(mark(X:S))
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A__F(s(0)) -> A__F(a__p(s(0)))
 A__F(s(0)) -> A__P(s(0))
 A__P(s(X:S)) -> MARK(X:S)
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(f(X:S)) -> A__F(mark(X:S))
 MARK(f(X:S)) -> MARK(X:S)
 MARK(p(X:S)) -> A__P(mark(X:S))
 MARK(p(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 a__f(0) -> cons(0,f(s(0)))
 a__f(s(0)) -> a__f(a__p(s(0)))
 a__f(X:S) -> f(X:S)
 a__p(s(X:S)) -> mark(X:S)
 a__p(X:S) -> p(X:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S)
 mark(f(X:S)) -> a__f(mark(X:S))
 mark(p(X:S)) -> a__p(mark(X:S))
 mark(s(X:S)) -> s(mark(X:S))

Problem 1: 

SCC Processor:
-> Pairs:
 A__F(s(0)) -> A__F(a__p(s(0)))
 A__F(s(0)) -> A__P(s(0))
 A__P(s(X:S)) -> MARK(X:S)
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(f(X:S)) -> A__F(mark(X:S))
 MARK(f(X:S)) -> MARK(X:S)
 MARK(p(X:S)) -> A__P(mark(X:S))
 MARK(p(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 a__f(0) -> cons(0,f(s(0)))
 a__f(s(0)) -> a__f(a__p(s(0)))
 a__f(X:S) -> f(X:S)
 a__p(s(X:S)) -> mark(X:S)
 a__p(X:S) -> p(X:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S)
 mark(f(X:S)) -> a__f(mark(X:S))
 mark(p(X:S)) -> a__p(mark(X:S))
 mark(s(X:S)) -> s(mark(X:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A__F(s(0)) -> A__F(a__p(s(0)))
 A__F(s(0)) -> A__P(s(0))
 A__P(s(X:S)) -> MARK(X:S)
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(f(X:S)) -> A__F(mark(X:S))
 MARK(f(X:S)) -> MARK(X:S)
 MARK(p(X:S)) -> A__P(mark(X:S))
 MARK(p(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 a__f(0) -> cons(0,f(s(0)))
 a__f(s(0)) -> a__f(a__p(s(0)))
 a__f(X:S) -> f(X:S)
 a__p(s(X:S)) -> mark(X:S)
 a__p(X:S) -> p(X:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S)
 mark(f(X:S)) -> a__f(mark(X:S))
 mark(p(X:S)) -> a__p(mark(X:S))
 mark(s(X:S)) -> s(mark(X:S))

Problem 1: 

Reduction Pairs Processor:
-> Pairs:
 A__F(s(0)) -> A__F(a__p(s(0)))
 A__F(s(0)) -> A__P(s(0))
 A__P(s(X:S)) -> MARK(X:S)
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(f(X:S)) -> A__F(mark(X:S))
 MARK(f(X:S)) -> MARK(X:S)
 MARK(p(X:S)) -> A__P(mark(X:S))
 MARK(p(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 a__f(0) -> cons(0,f(s(0)))
 a__f(s(0)) -> a__f(a__p(s(0)))
 a__f(X:S) -> f(X:S)
 a__p(s(X:S)) -> mark(X:S)
 a__p(X:S) -> p(X:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S)
 mark(f(X:S)) -> a__f(mark(X:S))
 mark(p(X:S)) -> a__p(mark(X:S))
 mark(s(X:S)) -> s(mark(X:S))
-> Usable rules:
 a__f(0) -> cons(0,f(s(0)))
 a__f(s(0)) -> a__f(a__p(s(0)))
 a__f(X:S) -> f(X:S)
 a__p(s(X:S)) -> mark(X:S)
 a__p(X:S) -> p(X:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S)
 mark(f(X:S)) -> a__f(mark(X:S))
 mark(p(X:S)) -> a__p(mark(X:S))
 mark(s(X:S)) -> s(mark(X:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a__f](X) = 2.X + 2
[a__p](X) = X
[mark](X) = 2.X
[0] = 2
[cons](X1,X2) = 2.X1 + 2
[f](X) = 2.X + 2
[fSNonEmpty] = 0
[p](X) = X
[s](X) = 2.X
[A__F](X) = 2.X + 2
[A__P](X) = X + 2
[MARK](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 A__F(s(0)) -> A__F(a__p(s(0)))
 A__P(s(X:S)) -> MARK(X:S)
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(f(X:S)) -> A__F(mark(X:S))
 MARK(f(X:S)) -> MARK(X:S)
 MARK(p(X:S)) -> A__P(mark(X:S))
 MARK(p(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 a__f(0) -> cons(0,f(s(0)))
 a__f(s(0)) -> a__f(a__p(s(0)))
 a__f(X:S) -> f(X:S)
 a__p(s(X:S)) -> mark(X:S)
 a__p(X:S) -> p(X:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S)
 mark(f(X:S)) -> a__f(mark(X:S))
 mark(p(X:S)) -> a__p(mark(X:S))
 mark(s(X:S)) -> s(mark(X:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A__F(s(0)) -> A__F(a__p(s(0)))
->->-> Rules:
 a__f(0) -> cons(0,f(s(0)))
 a__f(s(0)) -> a__f(a__p(s(0)))
 a__f(X:S) -> f(X:S)
 a__p(s(X:S)) -> mark(X:S)
 a__p(X:S) -> p(X:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S)
 mark(f(X:S)) -> a__f(mark(X:S))
 mark(p(X:S)) -> a__p(mark(X:S))
 mark(s(X:S)) -> s(mark(X:S))
->->Cycle:
->->-> Pairs:
 A__P(s(X:S)) -> MARK(X:S)
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(f(X:S)) -> MARK(X:S)
 MARK(p(X:S)) -> A__P(mark(X:S))
 MARK(p(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 a__f(0) -> cons(0,f(s(0)))
 a__f(s(0)) -> a__f(a__p(s(0)))
 a__f(X:S) -> f(X:S)
 a__p(s(X:S)) -> mark(X:S)
 a__p(X:S) -> p(X:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S)
 mark(f(X:S)) -> a__f(mark(X:S))
 mark(p(X:S)) -> a__p(mark(X:S))
 mark(s(X:S)) -> s(mark(X:S))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Reduction Pairs Processor:
-> Pairs:
 A__F(s(0)) -> A__F(a__p(s(0)))
-> Rules:
 a__f(0) -> cons(0,f(s(0)))
 a__f(s(0)) -> a__f(a__p(s(0)))
 a__f(X:S) -> f(X:S)
 a__p(s(X:S)) -> mark(X:S)
 a__p(X:S) -> p(X:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S)
 mark(f(X:S)) -> a__f(mark(X:S))
 mark(p(X:S)) -> a__p(mark(X:S))
 mark(s(X:S)) -> s(mark(X:S))
-> Usable rules:
 a__f(0) -> cons(0,f(s(0)))
 a__f(s(0)) -> a__f(a__p(s(0)))
 a__f(X:S) -> f(X:S)
 a__p(s(X:S)) -> mark(X:S)
 a__p(X:S) -> p(X:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S)
 mark(f(X:S)) -> a__f(mark(X:S))
 mark(p(X:S)) -> a__p(mark(X:S))
 mark(s(X:S)) -> s(mark(X:S))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a__f](X) = 0
[a__p](X) = 1/2.X + 1/2
[mark](X) = X
[0] = 2
[cons](X1,X2) = 2.X2
[f](X) = 0
[fSNonEmpty] = 0
[p](X) = 1/2.X + 1/2
[s](X) = 2.X + 1/2
[A__F](X) = 2.X
[A__P](X) = 0
[MARK](X) = 0

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a__f(0) -> cons(0,f(s(0)))
 a__f(s(0)) -> a__f(a__p(s(0)))
 a__f(X:S) -> f(X:S)
 a__p(s(X:S)) -> mark(X:S)
 a__p(X:S) -> p(X:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S)
 mark(f(X:S)) -> a__f(mark(X:S))
 mark(p(X:S)) -> a__p(mark(X:S))
 mark(s(X:S)) -> s(mark(X:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 A__P(s(X:S)) -> MARK(X:S)
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(f(X:S)) -> MARK(X:S)
 MARK(p(X:S)) -> A__P(mark(X:S))
 MARK(p(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 a__f(0) -> cons(0,f(s(0)))
 a__f(s(0)) -> a__f(a__p(s(0)))
 a__f(X:S) -> f(X:S)
 a__p(s(X:S)) -> mark(X:S)
 a__p(X:S) -> p(X:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S)
 mark(f(X:S)) -> a__f(mark(X:S))
 mark(p(X:S)) -> a__p(mark(X:S))
 mark(s(X:S)) -> s(mark(X:S))
-> Usable rules:
 a__f(0) -> cons(0,f(s(0)))
 a__f(s(0)) -> a__f(a__p(s(0)))
 a__f(X:S) -> f(X:S)
 a__p(s(X:S)) -> mark(X:S)
 a__p(X:S) -> p(X:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S)
 mark(f(X:S)) -> a__f(mark(X:S))
 mark(p(X:S)) -> a__p(mark(X:S))
 mark(s(X:S)) -> s(mark(X:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a__f](X) = 2.X + 2
[a__p](X) = X
[mark](X) = 2.X + 1
[0] = 0
[cons](X1,X2) = X1 + 1
[f](X) = 2.X + 2
[fSNonEmpty] = 0
[p](X) = X
[s](X) = 2.X + 2
[A__F](X) = 0
[A__P](X) = X + 1
[MARK](X) = 2.X + 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(f(X:S)) -> MARK(X:S)
 MARK(p(X:S)) -> A__P(mark(X:S))
 MARK(p(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 a__f(0) -> cons(0,f(s(0)))
 a__f(s(0)) -> a__f(a__p(s(0)))
 a__f(X:S) -> f(X:S)
 a__p(s(X:S)) -> mark(X:S)
 a__p(X:S) -> p(X:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S)
 mark(f(X:S)) -> a__f(mark(X:S))
 mark(p(X:S)) -> a__p(mark(X:S))
 mark(s(X:S)) -> s(mark(X:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(f(X:S)) -> MARK(X:S)
 MARK(p(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 a__f(0) -> cons(0,f(s(0)))
 a__f(s(0)) -> a__f(a__p(s(0)))
 a__f(X:S) -> f(X:S)
 a__p(s(X:S)) -> mark(X:S)
 a__p(X:S) -> p(X:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S)
 mark(f(X:S)) -> a__f(mark(X:S))
 mark(p(X:S)) -> a__p(mark(X:S))
 mark(s(X:S)) -> s(mark(X:S))

Problem 1.2: 

Subterm Processor:
-> Pairs:
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(f(X:S)) -> MARK(X:S)
 MARK(p(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 a__f(0) -> cons(0,f(s(0)))
 a__f(s(0)) -> a__f(a__p(s(0)))
 a__f(X:S) -> f(X:S)
 a__p(s(X:S)) -> mark(X:S)
 a__p(X:S) -> p(X:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S)
 mark(f(X:S)) -> a__f(mark(X:S))
 mark(p(X:S)) -> a__p(mark(X:S))
 mark(s(X:S)) -> s(mark(X:S))
->Projection:
 pi(MARK) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a__f(0) -> cons(0,f(s(0)))
 a__f(s(0)) -> a__f(a__p(s(0)))
 a__f(X:S) -> f(X:S)
 a__p(s(X:S)) -> mark(X:S)
 a__p(X:S) -> p(X:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S)
 mark(f(X:S)) -> a__f(mark(X:S))
 mark(p(X:S)) -> a__p(mark(X:S))
 mark(s(X:S)) -> s(mark(X:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.